Exploring the legacy of the rationalist medieval Torah scholars, and various other notes

Thursday, March 14, 2013

Happy Pi Day

(A re-port from two years ago, with some updates)

Today, March 14th, is official Pi Day. (The date is chosen because it's 3.14.) You can check out this link for a brief history of Pi. Meanwhile, I'd like to share a few thoughts about Pi in Jewish thought. I have not been able to research this topic anywhere near as thoroughly as I usually do, but I don't want to miss the date, so here are some thoughts.

There are those who claim that the description of King Shlomo's Pool having a diameter of 10 cubits and a circumference of 30 cubits is an example of a scientific error in the Bible. Now, I am certainly not ideologically closed to the idea of the Bible being scientifically inaccurate - there are several examples of this, such as with the kidneys, dew, firmament, etc., for which we invoke the concept of Dibra Torah k'lashon bnei Adam. However, I don't believe that King Shlomo's Pool is an example of this; it's just a convenient way of describing it. Although, it is perhaps a little problematic here because instead of saying that its circumference was 30 cubits, it says that a thirty-cubit line could encircle it.

A few days ago, a reader who is convinced that Chazal had a supernatural source of knowledge about the natural world gave King Shlomo's Pool as an example. Here is the idea, as presented by Rav Mordechai Kornfeld (and follow the link for further discussion):

A fascinating insight regarding the value of pi is attributed to the Vilna Ga'on. (Actually, there is no source to substantiate the claim that the Vilna Ga'on said it. The actual source for the insight may be credited to Matityahu ha'Kohen Munk (Frankfurt-London), who published the thought in the journals "Sinai," Tamuz 1962, and "ha'Darom," 1967.) In the verse that the Gemara cites as the source for the ratio of the circumference to the diameter (Melachim I 7:23), there is a "Kri" and a "Kesiv" -- a word that is pronounced differently than it is spelled. The word in the verse is written "v'Kaveh" (with the letter "Heh" at the end), but it is pronounced "v'Kav" (with no "Heh" at the end). The Gematriya of the word "Kav" is 106, and the Gematriya of the word "Kaveh" is 111. The ratio of the Kesiv (111) to the Kri (106), or 111/106, is 1.0471698. This value represents the ratio of the value for pi to 3 (3.1415094/3 = 1.0471698).

The question is, does this provide evidence for Chazal having supernatural sources of knowledge? I don't think so, for several reasons.

First of all, a person could argue that the kav/kaveh curio is simply a coincidence. It's not a matter of something being accurate to seven decimal places. There are two numbers, 106 and 111, which can be manipulated to give a certain value. There are doubtless plenty of two and three digit numbers which can be manipulated to give a similar value, and there are plenty of two and three digit numbers that can be derived from a verse. Some will see this as unduly skeptical, and at the moment, I am inclined to agree, since it's just too neat that it's exactly with the word describing the circumference that this gematria is found. But I don't think that I can conclusively show that it's not a coincidence.

As for the significance of the kri/ksiv, while Malbim and (of course) Maharal ascribe significance to both kri and ksiv, according to Radak they simply reflect uncertainties that arose in transmission.

Then, even if one wants to claim that Pi is encoded in kav/kaveh, does this reflect a supernatural encoder? The value of Pi was known in ancient times to several decimal places, and a human could encode it in this way. There is a Greek Pythagorean motto "God is ever a geometer" (ἀεὶ ὁ Θεὸς ὁ μέγας γεωμετρεῖ) — the number of letters in each of the six words are the first six digits of pi. A cute and deliberately constructed device, but not one that indicates that the composer of either the phrase or the language was supernatural!

Finally, even if one does feel that this strongly points to a supernatural encoder, it is not evidence of Chazal possessing a supernatural source of knowledge. The verse is assumed to have been written with Divine Inspiration, which means that God has supernatural knowledge, not man. With regard to Chazal, it does not appear that they knew the value of Pi to any decimal places. The Gemara gives the value of Pi as being 3 (Eruvin 14a), and Tosafos points out that, based on the context, the Gemara does not seem to be giving an approximation. Of course, there are various apologetics which argue otherwise, but Tosafos apparently didn't find them convincing. Thus, if someone wants to believe that the Gemara did not mean this, they can do so, but one cannot use the topic of Pi to prove that Chazal had superior knowledge of the natural world.

Furthermore, the Mishnah (Ohalos 12:6) says that "A square is greater than a circle by one-fourth," referring to the perimeter of each when the circle is drawn to the height of the square. This is true if Pi is assumed to be 3, but given a more accurate value of Pi, the perimeter of the square is actually closer to one-fifth longer than that of the circle.

Some readers will doubtless find it hard to accept that Chazal believed Pi to be 3. The question is whether there is basis for their disbelief, and an analysis of the Gemara and Rishonim reveals that there were much more basic mathematical errors committed by some (but not all) of Chazal. Tosafos (Eruvin 76a) says that Rabbi Yochanan and the Gemara in Sukkah misunderstood a statement by the judges of Caesarea to mean that the diagonal of a square is equal to twice the length of its side. Tosafos states that Rabbi Yochanan subscribed to this understanding of the judges of Caesarea, and that the Gemara in Sukkah rejected it precisely because it is mathematically inaccurate. Rashba expresses surprise at Tosafos attributing a simple mathematical error to Chazal, and he gives an alternate explanation, but he does not deny that Tosafos does indeed say this! Ran likewise expresses surprise that the judges of Caesarea erred in a simple mathematical matter, and cites an alternate explanation of Rabbi Yochanan’s misunderstanding of what the judges of Caesarea were saying, which somewhat lessens the error, but still leaves Rabbi Yochanan making genuine errors of both interpretation and mathematics. Tosafos HaRosh states similarly. Given all this, there is no reason not to take the Gemara's statement about the ratio of a circle's circumference to its diameter at face value.

Finally, we have Rambam on record as being the first person in recorded history to explicitly describe Pi as being an irrational number (see Wired Magazine's article on this). I don't know whether it is amusing or sad that some people co-opt the Rambam for anti-rationalist purposes. Jonathan Rosenblum declared that Rambam's statement about Pi is evidence that Torah scholars have supernatural sources of knowledge about the natural world. But first of all, while Rambam was the first to write this explicitly, it had already been hinted at by earlier Greek writers. Secondly, the idea that Rambam knew this via kabbalah or some other such source is ludicrous and a distortion of Rambam's fundamental ideology. Rambam himself wrote that even Chazal had no such supernatural sources of knowledge; he certainly did not consider himself to be privy to kabbalistic secrets!

Have a happy Pi day, and let's not undermine the credibility of Torah and Judaism by making extreme claims that do not stand up to scrutiny. There's enough to be proud of in our religion without having to resort to such shtick!

53 comments:

Actually wouldn't you say it's very clever: the very word(s) to describe the circumference "kav...yasov", the way it is said represents the index of the way PI is always "said" in Chazal - 3, whereas the written but unspoken, or "underlying" Gematria represents the index of PI "below the surface"? KAVA would be gramtically incorrect, and in Divrey Hayamim 2: 4:2 it says KAV without the alternative reading so they obviously knew the correct reading. Come on, there's something going on here...

In divray hayamim it says the SAME passuk with just KAV so "they" should have just corrected it, and it's not really an accidental inaccuracy but rather a purposeful one, because a KAV/KAVA KERI/KETIV is also done in Zecharia 1:16 in a different context, and it is referring to a measuring tool (the mefarshim don't agree on what it's measuring exactly - but you could make the same allusion to the common measuring methods and the accurate ones). So what you're saying, Chaim, does not seem to be plausible.

I've always thought that this was a fascinating calculation and - even though I'm heretical in my beliefs - still don't think that the k'ri k'siv ratio is coincidence. But, as you say, it isn't proof of divine knowledge.

Did chazal believe that pi=3? A very simple measurement of a circle with a string will show that it isn't. It is much more likely that a whole number was used because it is immensely more practical (unless, of course, one needs a more precise value for - say - architecture, something that chazal was not concerned with.)

But forget the Greeks; this is historically too late re Shlomo's pool. A quick wiki on the history of pi says that "the earliest known textually evidenced approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value."

Well, although I like the drash and I hate to diminish it, the math is wrong:

Pi is 3.141592653589793.....NOT 3.1415094 like the e-mail says

So we have:111/106 = 1.047169811320755...pi/3 = 1.047197551196598...

So it is accurate to 5 sig-figs not 8.

Even so, it could be that 5 sig-figs is enough for almost any purpose, i.e. if we take the 111/106 ratio to find an approximate value to pi we get:333/106 = 3.141509433962265...which is a 99.997% accurate approximation of PI which is pretty darn close!

It amazes me that on PI-day nobody noticed the e-mail was using the wrong value for pi! :)

Are there any Jewish commentators who try to resolve the dimensions of King Shlomo's pool in the same way some Christians do, by referring to the inner and outer diameters?http://www.bibleprobe.com/pi.htm

Reuven Meir: Note the superfluous hei in v'kav. Perhaps it somehow alludes to taking the ratio of v’kav and v'chameish - the value of hei. If you do so, it just so happens that the ratio of the numerical values of v'chameish (355) and v'kav (113) [including one for the word itself] equals 3.14159292...which is precisely six decimal places of accuracy – one part in a million of pi's exact value, 0.00000026 more than pi, and 99.9999915...% similar to pi. It turns out that there is no better approximation for pi as a ratio of the numerical values of any two words, or of any other whole numbers less than ten-thousand. This calculation is even internally consistent due to the fact that it is veiled within a pasuk containing the ratio of circumference to diameter.

[edit] Biblical valueIt is often claimed that the Bible states that π is exactly 3, based on a passage in 1 Kings 7:23 (ca. 971–852 BCE) and 2 Chronicles 4:2 giving measurements for the round basin located in front of the Temple in Jerusalem as having a diameter of 10 cubits and a circumference of 30 cubits. Rabbi Nehemiah explained this in his Mishnat ha-Middot (the earliest known Hebrew text on geometry, ca. 150 CE) by saying that the diameter was measured from the outside rim while the circumference was measured along the inner rim. This interpretation implies a brim 0.22535 cubit (or, assuming an 18-inch “cubit”, some 4 inches) thick, or roughly one “handbreadth” (cf. 1 Kings 7:24 and 2 Chronicles 4:3).

The interpretation of the biblical passage is still disputed[13][14], however, and other explanations have been offered, including that the measurements are given in round numbers (as the Hebrews tended to round off measurements to whole numbers[citation needed]), or that cubits were not exact units, or that the basin may not have been exactly circular, or that the brim was wider than the bowl itself. Many reconstructions of the basin show a wider brim (or flared lip) extending outward from the bowl itself by several inches.[15]

The issue is discussed in the Talmud and in Rabbinic literature.[16] Among the many explanations and comments are these:

In 1 Kings 7:23 the word translated 'measuring line' appears in the Hebrew text spelled QWH, but elsewhere the word is most usually spelled QW. The ratio of the numerical values of these Hebrew spellings is 111⁄106. If the putative value of 3 is multiplied by this ratio, one obtains 333⁄106 = 3.141509433... – within 1/10,000th of the true value of π, a convergent for π which is more accurate than 22⁄7, although not nearly as good as the next one 355⁄113. Maimonides states (ca. 1168 CE) that π can only be known approximately, so the value 3 was given as accurate enough for religious purposes. This is taken by some[17] as the earliest assertion that π is irrational.

There's an additional problem with the kri/ksiv argument. Taken to its logical conclusion it would imply that the author thought that that slightly more accurate number was the exact value of Pi which is of course wrong. So if one really believed that argument, it would potentially come out to an argument against the divine inspiration of the text.

In contrast, the verse using a simple approximation (or using a rope that stretched slightly or both) are much easier.

Incidentally, the claim about the Greek phrase is particularly interesting- the decimal system is very much a modern innovation which is also arbitrary. To a culture using any other base, that phrase would have zero significance.

(Somewhat related, I was under the impression that there's a Gemarrah that uses the 22/7 value of Pi, but I don't know where it is and you don't mention it here. Am I misinformed?)

Yossi, to answer your question, no Chazal did not have decimal points. That wasn't invented to around 500 years ago. But rational numbers have been well-understood for a very long time.

Explicit use of continued fractions (http://en.wikipedia.org/wiki/Continued_fraction ) occurred around 500 c.e. or so although when it came to the West isn't clear. But you don't need continued fractions to do rational approximation. It just helps it make much more sense (the theory of continued fractions explains why 22/7 is a surprisingly good approximation for such a small denominator. 22/7 = 3.14285... as opposed to Pi= 3.14159... so the difference is about 1/790 which is much smaller than 1/7 (which is about where you would very naively expect the approximation to be).

Moreover, mathematicians in Greece and Babylon were working with approximations to Pi very early on without any use of decimals. And the same goes for the square root of 2 (which is implicitly what is being used in the issue of the diagonal of a square mentioned by Rabbi Slifkin in the original post). There's demonstrably no issue of a lack of techniques at the time since other people were able to work with them fine.

"There are doubtless plenty of two and three digit numbers which can be manipulated to give a similar value...But I don't think that I can conclusively show that it's not a coincidence."

I was inclined to believe the same thing- that the 111:106 was far from unique, so I did some programming to show that "There are doubtless plenty of two and three digit numbers..."

I wrote a little program to generate pairs of numbers that produce ratios close to PI:3. Starting at 1 and continuing, I took note of the best ratios as they appeared.The first pair, is of course 1/1 which gives the ratio of 1:1 which approximates pi=3. The next pair that produces a better result is 12/11 which gives us pi=3.2727. The next is 13/12, followed by 14/13 and so forth- until 22/21 which provides the approximation 3.1428. No better pair appears until 67/64, then 89/85, then 111/106- our value here. Until 1000, there are only two more pairs that give us better results: 244/233 and 355/339. (The latter number was alluded to in Simcha's comment.) And that's the best you can get for pairs under 10,000. It means that 111:106 is the third best ratio for numbers under 10,000. And it's the best that could be generated from a single letter difference- i.e. a kri/kesiv.

Coincidence? Perhaps, but it far less coincidental than the skeptic would have it.

I wrote: "Are there any Jewish commentators who try to resolve the dimensions of King Shlomo's pool in the same way some Christians do, by referring to the outer and inner diameters?"

Shimon writes: "Rabbi Nehemiah explained this in his Mishnat ha-Middot (the earliest known Hebrew text on geometry, ca. 150 CE) by saying that the diameter was measured from the outside rim while the circumference was measured along the inner rim."

"The Ra'avad in his hakdomoh to sefer yetzirah believes that pi is 3.2."

Does he 'believe' it to be that value exactly? I find it hard to believe that someone who would know what pi is, would not know that 3.2 is just an approximation. More likely, he gave the value to the nearest fifth. Just as we measure many things to the closest quarter inch....

If this discussion of a "sea", or large bowl, had been referring to what is called an "ideal" bowl (a mathematical object, not existing in a physical sense, and having no thickness that could be felt or handled), then the text would indeed be claiming that the value of pi is three. But the text is referring to a real-world physical object, having the thick sidewalls necessary to support its own weight.Now that you know how to measure cubits, can you see that it would be rather difficult to measure the curved surface of a bowl in cubits? Instead, a straightened rope would be used to measure the length. The rope would then have been moved to outline a circle with the desired circumference. Also, Hiram would not have just tossed some brass in the furnace and waited to see what came out. He would have designed the piece and would have given his workmen instructions.To make a "sea" like this would likely have required a mold. The outer mold would have one dimension, and the inner mold would have another. Hiram would have given his workmen instructions regarding these measurements.Now that you have some background information, let's look at the numbers:

The CalculationsHere again is the quote being referred to:"And he [Hiram] made a molten sea, ten cubits from the one rim to the other it was round all about, and...a line of thirty cubits did compass it round about....And it was an hand breadth thick...." — First Kings, chapter 7, verses 23 and 26The bowl is said to have had a circumference of thirty cubits and a diameter of ten cubits. The diameter is said to be "from one rim to the other", so this would be the outer diameter; that is, the diameter of the outer mold used to make the bowl.

The circumference is not specified as being the inner or outer circumference, but since using the outer circumference would give us the "ideal" bowl (with no width or thickness), let's instead use the inner circumference, which also, reasonably, would have been the circumference of the mold used to form the inside of the bowl. That is, we will use the two measurements which were necessary for the casting of the piece.

Using eighteen inches for one cubit, we have the following:outer diameter: 10 cubits, or 180 inches outer radius: 5 cubits, or 90 inches inner circumference: 30 cubits, or 540 inches To find the "Jewish" or "Bible" value for pi, we need to have the inner radius. Once we have that value, we can plug it into the formula for the circumference and compare with the given circumference value of 540 inches.Since the thickness of the bowl is given as one handbreadth, then the inner radius must be:90 – 4 = 86 inchesLet's do the calculations:inner radius: 86 inches inner circumference: 540 inches

The circumference formula is C = 2(pi)r, which gives us:540 = 2(pi)(86) 540 = 172(pi) Solving, we get pi = 540/172 = 135/43 = 3.1395348837..., or about 3.14.Um... Isn't "3.14" the approximation we all use for pi? Hmm.... I guess the Torah was fairly accurate after all.

Based on an article by Stapel, Elizabeth, "The 'Jewish' or 'Bible' Value of 'pi'"

See David Garber and Boaz Tzaban in HISTORIA MATHEMATICA 25 (1998), 75–84ARTICLE NO. HM972185 for three three explanations to the accuracy of the biblical pi.

This is off topic but realted to the issue of Rambam mistakenly being thought of as a secret Kabbalist. Here is a comprehensive historical overview of the issue: http://louisjacobs.org/articles/view.php?id=94.

The text of Melahim speaks of the semi-spherical bowl, with outer diameter of 10 cubits, outer radius 5 cubits and inner circumference of 30 cubits. This results in outer circumference of the bowl 10Pi or about 31.1 cubits.

So the exact value of Pi was most certainly known back then.

I think a lot of people get confused because the text uses outer diameter and INNER circumference to describe the bowl, which is not conventional from the modern point of view.

There is a Greek Pythagorean motto "God is ever a geometer" (ἀεὶ ὁ Θεὸς ὁ μέγας γεωμετρεῖ) — the number of letters in each of the six words are the first six digits of pi. A cute and deliberately constructed device, but not one that indicates that the composer of either the phrase or the language was supernatural!

Device? for what? That number of letters conceit looks at pi in a positional base 10 system. Wikipedia FWIW states that the Babylonians used a positional base 60 system, so reading pi into the the Pythagorean motto wouldn't have worked there, and while the Greeks did indeed use a decimal system, I don't think it was positional.

The sages took the value of pi (the ratio of the circumfrence of a circle to its diameter) as 3 in many places in the talmud. Their conjectures about the brazen water reservoir of Shlomo in T.B. Eruvin 14a had to do with the understandable assumption that the 10 ama diameter of the round vessel that is mentioned in the cited verse was the inner diameter (only the i.d. is of interest in a volume determination), while the stated 30 ama circumference was the outer dimension (the only one easily measured with a measuring line). They speculate that the vessel had a fine edge like a chalice so that the o.d. and i.d. were very nearly the same. The objection is still raised that there remains a difference, and the answer given is that the stated circumference of the vessel is the inner one. The Tosafot conclude from this evident understanding of the gemara that pi was assumed to be exactly 3.

I am not excited with the kav/kava correction ratio that converts 3 (30/10) into a better approximation for pi. It is a bit of cute numerology derived on an ad-hoc basis. As has been pointed out 355/113 gives a decidedly better approximation. If one wishes to become 'mystical' about the matter, consider an arithmetic series of pairs of odd numbers, i.e., 113355. If one takes the 1st 4 digits of the series (1133)and divides the latter pair by the former, one gets 3 (the approximation used by the sages). If one includes the next pair (55) and divides the 2nd set of 3 (355) by the 1st (113), one gets the best approximation using numbers below 1000.

The outstanding question about R' Yochanan's halacha in T.B. Eruvin 76a is that he appears to assume that the diagonal of a square is given by the sum of the sides. Such an assumption is implicit in the idea that a circular opening in a wall must have a circumference of 24 tefachim in order to enclose a square space of 4x4 tefachim. If the diagonal is the sum of the sides of the square, i.e., 8 tefachim and pi is 3 then the circumference of the round opening is 24. This understanding also rationalizes his stipulation that the bottom of the circular opening must be within 8 tefachim of the ground in order for the bottom of the enclosed square opening to be within 10 tefachim of the ground. The problem with this understanding of R' Yochanan is that it is not only mathematically and logically wrong (a straight line distance between 2 points on a plane must be less than any broken path), but that a much better approximation for the diagonal of a square relative to its side was used by other sages, i.e., 1.4 or 7/5 - not 2. This value of 1.4 for sqrt 2 (1.414..) is only 1% off the accurate value while using 3 for pi is 4.5% off.

Tosafot in Eruv. 76b not only allude to this error, but they also explain that the citation from the sages of Ceasaria that a circle is 1.5 times the enclosed square (assuming that pi = 3) refers to the areas - not the perimeters. Indeed, the ratio of the areas is 1.5 (for pi=3). They also beautifully demonstrate how to calculate the area of circle.

Rabbi Slifkin, since you are updating this, you should probably heed Reuven Meir's comment. By using the words "being accurate to seven decimal places" you seem to be implying that 111/106 gives pi to 7 decimal places, when in reality it is only 4. Not a big deal, but you may as well fix it.

Hi just saw your piece on locusts.I fail to understand something; you see by the bris of your son you wrote the reason you did metziza (eventhough we are aware today that there are no medical benefits) is because in your words "why tamper with a minhag"Yet Ashkenazim have more than just a minhag not to eat locusts yet you happily disregard it for the reasons you stated.The way you pick and choose is an embarassment to mature and broad minded jews.As a Jewish Grammar educated boy one would have expected to see more consistancy from you.

No, they have LESS than a minhag. They just lack a tradition of eating them; they do not have a tradition NOT to eat them. You might as well say that Ashkenazim have a minhag not to live in Eretz Yisrael.

You seem to think that I am some sort of maverick in this regard. In fact, many Ashkenazi poskim say that it is perfectly legitimate to accept the North African mesorah on locusts.

The way you pick and choose

Everybody picks and chooses. Some just do it with more analysis, consistency and self-awareness than others.

As was pointed out above:p (Pi) – is the ratio of the circumference of a circle to its diameter. It is an irrational number and cannot be expressed as an exact fraction. p is ~= 3.14159265358979 accurate to 15 decimal digits. The closest approximation in a one-digit fraction is 1 divided into 3 = 3, which is used by the Gemara and is accurate to 1 decimal digit. The closest approximation with a two-digit fraction is 7 divided into 22 ~= 3.142857, which is accurate to 3 decimal digits and is also used by Chazal . The closest approximation with a three-digit fraction is 113 divided into 355 ~= 3.141592920 which is accurate to 7 decimal digits.

However, Gemara in Succah 8b and Eruvin 76b both quote a ma’amar from Rabbi Yochanan that a 4 by 4 square can be circumscribed by a circle that has an hakayfoh of 24. The Gemara then brings proof from the Rabbis of Caesarea who state, “a circle inscribed in a square, is one-fourth; a square which is inscribed in a circle, is one-half”. The normal meaning of hakayfoh is circumference or perimeter. This leads to a large difference between the 24 of R’ Yochanan and the 16.8 the Gemara calculates (or even the 17.77 with more accurate calculations).

Based on the Me'iri, who says so explicitly, and Tosafos and the Ritva, from whom we can derive it, Rabbi Yochanan is talking about area and is correct in both Succah and Eruvin. Rabbi Yochanan's statement that “the circumference of the Sukah must be large enough to seat 24 people in it” does not mean that the circumference must be 24 Amos, but that there must be room for 24 people occupying 24 square Amos inside the circumference -- in other words, the area of the circle must be 24 square Amos!

The Rabbis of Ceasarea are then brought as proof and state that the area of a circle that is drawn around a square which is 4 by 4 is calculated by subtracting ¼ of the area of the circumscribing square or adding ½ of the area of the inscribed square and is exactly equal to 24 square Amos.

This is what Rabbi Yochanan meant when he said that the circle must have within its circumference an area of 24 in both Succah and Eruvin.

Further when Rabbi Yochanan states in Eruvin that in order to get the inscribed square of 4 by 4 Tefachim below a height of 10 Tefachim, at least 2 Tefachim and a bit of the circular window must be below ten Tefachim; he is talking about the area of segment CBL which is exactly 2 square tefachim . If 2 square tefachim and a bit of the circle are below 10 tefachim, the bottom of the 4 by 4 square will therefore be below 10 tefachim and the Pesach (opening) is valid, and allows the Chatzeros to be joined in one Eruv.

Thus if we view the statements of Rabbi Yochanan and the Rabbis of Caesarea as referring to area and with the value of p equal to 3, they are exactly correct.

Even in plain geometry we find Chazal determining laws based on homiletics, and only afterwards trying to make the facts fit these laws. In Tractate Eiruvin 14a the Talmud says:

"Anything which has, in its circumference, 3 tefachs, has one tefach in diameter. How do we know this? Rabbi Jochanan said, it is written in the Scripture: 'And he [Solomon] made a molten sea, ten amahs from one brim to the other. It was round all about, and its height was five amahs. And a line of thirty amahs circled it' (I Kings 7:23)."

The Talmud rules that the ratio between a circle's circumference and its radius, known as pi, is 3. In fact, this number is irrational (impossible to represent as a finite common or decimal fraction), and taken to 10 decimal places, pi=3.1415926536.

One might say that Chazal also knew that true pi is more than 3 and only tried to find a Halachically valid approximation of this number -- but this is impossible because of the Gemara in Bava Batra 14b:

"And if you think about the Torah Scroll [of the Temple] which had 6 tefachs in circumference, provided that everything that has 3 tefachs in circumference has one tefach in diameter and provided that the Torah scroll was rolled to its middle [i.e. it was rolled on two wooden shafts like our Torah scrolls are], we have more than 2 tefachs between one handle and another -- so how could it enter the 2 tefachs of free space [in the Holy Ark]? Rav Acha the son of Jacob said: the Torah scroll of the Temple was rolled to its beginning [i. e. it was rolled on one wooden shaft only]. And yet, since it was 2 tefachs in diameter, how could it enter 2 tefachs of free space [in the Ark]? Rav Ashei said: they did not wind all the Torah scroll on the pivot, but left a part of it unwound, put the scroll into the Ark, and then folded the remaining part of the scroll onto it."

They thought a Torah scroll 6 tefachs in circumference to be exactly 2 tefachs in diameter, so they considered it to be practically impossible to put such a scroll into a space of exactly 2 tefachs, unless one does not wind all the parchment of the scroll on its wooden shaft, thus leaving some free space to adjust the scroll in the Ark. Only after he puts the scroll into the Ark does he folds the remaining parchment and put it above the scroll.

Of course, were the Sages aware of the real value of pi -- or at least of the approximation 22/7 known to ancient Greeks centuries before the Talmudic era, they would have understood that the real diameter of a scroll 6 tefachs in circumference is about 1.9 tefachs and that nobody would need any special tricks to put it into 2 tefachs of free space. It is not difficult to determine that pi is significantly more than 3. All one needs is a ruler and a measuring rope. Nonetheless Chazal preferred to determine reality from verses and law instead of basing law on reality.

Avraham, while the statement of R' Yochanan in Sukkah 8b that a minimum round sukkah must fit 24 people (according to the Tannah that a minimum sukkah must be 4x4 amot) could well be referring to area - as opposed to the gemara's interpretation of R' Yochanan's position, that is not possible in Eruvin. In Eruvin 76a, As you cited, R' Yochanan refers to the hekaif (circumference) of a minimum round opening in the dividing wall of adjoining yards. That hekeif must be 24 tefachim in his view. Had he been referring to the area - a measurement that could not have been easily made, he would have used a term like 'shetach'. Moreover, how do you get an area of 2 sq. tefachim below the inscribed 4x4 tefach square? The difference in areas between the circle and inscribed square is (pi-2)/2. If pi is taken as 3, the total area outside the square is 1/2. The bottom segment of the 'excess' area is then 1/8 - not 2. However, if R' Yochanan's circle has a circumference of 24 based on his assumed square diagonal of 8, then the vertical diameter of the circle is 8. Since the enclosed square has a height of 4, that leaves 2 tefachim on the bottom (and 2 on top), i.e., this understanding of R' Yochanan in Eruvin is both consistent with the language used and with his ostensible 'mathematics'.

Why do I bother? The last poster may not have been serious, but shows a confusion between radius (R) and diameter (D). Pi is, precisely defined as the ratio of a circle's circumference (C) to its diameter (D). 2 pi, or tau - if you wish is the ratio of C to R. One definition is no more fundamental than the other. The area of a circle is given by pi*R^2 or (pi*D^2)/4, which is simpler than the corresponding expression for tau. While it is true that there are more equations in physics that contain 2 pi, that doesn't mean that 2 pi is mathematically more significant.

”… could well be referring to area - as opposed to the gemara's interpretation of R' Yochanan's position, that is not possible in Eruvin. … he would have used a term like 'shetach' … Since the enclosed square has a height of 4, that leaves 2 tefachim on the bottom …”

The principle, and math, in both Sukkah and Eruvin is the same proportionately. Thus the idea of 24, 1 by 1 units of measure works for both amos and tefachim. and the stalemates of the Rabbis of Caesarea is independent of the units.

I am not a linguistic expert, but I believe there was no word for area in R’ Yochanan’s time. Shetach came later. Area was indirectly referred to. E.g. bais kur an area covered by a kur of seed. So it would be natural to describe an area as the number of square units that would fit in a circle.

The 2 tefachim on the bottom are referring to the circle. Thus either the length of the line from the bottom of the circle to the bottom of the square (.8) or the area of the segment below the square which, assuming pi is 3, would be ¼ of the area outside the 4x4 square and inside the circle, exactly 2 (or using a more accurate pi, 2.28).

Avraham, yasher koach for your review of the commentators on the issue of a minimal round opening or succah and your effort to rationalize the view of R' Yochanan. However, the expression used by R' Yochanan in Eruvin is "chalon agul tzarich sheyehei behekeifo esrim varba'a tefachim [a round opening (that is just sufficient to negate the effect of a wall separating 2 yards) must be 24 tefachim). Hekeifo means circumference - not area, as can be seen in the gemara's question immediately following the above statement, "michdei, kol sheyeish behekeifo shlosho tefachim yesh bo berochbo tefach" (let's see, if something has a circumference of 3 tefachim, it has a diameter of 1). The latter expression is used often in shas. While shetach may be a later expression for area, makom is quite ancient. R' Yochanan does not state "chalon agul tzarich sheyehei b'mkomo esrim ve'arba'ah" (a round opening must have a minimal area of 24 sq. tefachim).

Moreover, despite my prior concession that R' Yochanan in Succa may have been referring to area, his language indicates otherwise. He states, a succah made like an oven (i.e., a cylindrical succah) must be able to accomodate 24 people to be kosher. If he meant in the interior of the succah, it would hold 16 people each taking up 1x1 ama, i.e. in the enclosed 4x4 ama square (the round caps representing the areas outside the enclosed square have a maximum depth of just over 0.8 ama). The gemara concludes that he meant 24 people sitting around the perimeter, i.e., a circumference of 24. Your sincere attempt to justify R' Yochanan also leaves the gemara in Eruvin and Succah with a mistaken understanding of both R' Yochanan and the sages of Caesaria. So, either R' Yochanan is wrong or the gemara is wrong (the Tosafot conclude that both were mistaken).Incidentally, I was mistaken when I stated previously that the area outside the square is 1/2 (taking pi as 3). It is actually S^2/2, where S is the side of the square, i.e., 8, for S=4. The bottom 'cap' then has an area of 2 sq. tefachim - as you stated.

“… your effort to rationalize the view of R' Yochanan … So, either R' Yochanan is wrong or the gemara is wrong (the Tosafot conclude that both were mistaken).”

I can’t take the credit for the rationalization the Me'iri preceded me by a few centuries.

If you read my paper (linked above) you will see I am carful to not assign right or wrong, I merely go through the geometry with the gemara’s and modern math assumptions. I agree with you that the language brought by the gemara is circumference but then R’ Yochanan and the Rabbis of Caesarea are very wrong. If we assume R’ Yochanan and the Rabbis of Caesarea are talking about area, with the normal assumption of pi = 3, they are exactly correct in both Succa and Eruvin. That appears to leave the gemara with a bad quote from R’ Yochanan and then, based on the text they had, they then try very hard to get close to the reality (but don’t not quite make it, see appendix B of my paper).

I'll leave it as an exercise for the reader whether my Diameter-vs.-Radius confusion was accidental or subversively demonstrative of the problem with Pi...

Really, the mathematically and physically meaningful circle constant is the ratio of Circumference to Radius, Tau, or 2*pi. The number 2*pi, 6.28... shows up frequently in "real life" such as the reduced Planck constant and the Fourier transform.

Yes, you can still do the arithmetic with Pi, but as some Tau-ists put it, that's the equivalent of using the symbol "H" for One-Half, using 2*H instead of using the number One as the Muliplicative Identity, and celebrating how the special number H shows up in the Amazing Formula (any number) * (2*H) = (any number)

This is Rationalist-Jewishly very important. Realizing that Tau, 2*pi, is what "counts," we rightly attribute our sages' discussion of Pi rather than Tau to their adoption of the best math and science of their surrounding societies.

To find "special" non-rationalist meaning in Pi is to, chas vashalom, accuse the Creator of misleading His people by sending "ruach-hakodesh" hints backing the wrong mathematical horse!

Avraham, I have read your paper and complimented the effort. However, I disagree with your interpretation (or the Meiri) about R' Yochanan's statements on circular measurements being about area rather than circumference. As you admit, the language he used in Eruvin (hekeifa) alludes to circumference. Moreover, there was no direct way to measure circular areas in talmudic times. Finally, any error is attributable to R' Yochanan - not the sages of Caesaria since the latter were, indeed, referring to circular vs square areas in giving their rules for the ratios. The latter point is made by the Tosafot in Eruvin 76b(s.v. Rabbe Yochanan)

First of all, the definition of Pi as the ratio of circumference to diameter of a circle is only one of many, many possible definitions of Pi. Pi is everywhere in pure and applied mathematics, as is the constant e, the natural base of logarithms.

Euler's formula, called by Richard Feynman the jewel of mathematics, has Pi not 2Pi:

e^(i*Pi) = -1

It has e, Pi, the imaginary number i = sqrt(-1), and -1. That's a lot of important numbers in mathematics!

"A square is greater than a circle by one-fourth," This indicates that if the circle is 1, then the square is 1 plus 1/4 or a ratio of 1 is to 1.25. The Babylonians have used the same ratio for deriving their Pi value of 3.125.

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This website is an exploration into the rationalist approach to Judaism that was most famously presented by Maimonides. It will also explore contemporary rationalist approaches, as well as being a forum for various other notes. Well-written comments in the spirit of this enterprise will be posted; please include a name (even a pseudonym).